EXAM I, PHYSICS 4304



FINAL EXAM, PHYSICS 5305, May 2, 2009, Dr. Charles W. Myles

INSTRUCTIONS: Please read ALL of these before doing anything else!!!

1. PLEASE write on one side of the paper only!! It wastes paper, but it makes my grading easier!

2. PLEASE don’t write on the exam sheets, there is no room! If you don’t have paper, I’ll give you some.

3. PLEASE show ALL work, writing down at least the essential steps in the problem solution. Partial credit will be liberal, provided that essential work is shown. Organized work, in a logical, easy to follow order will receive more credit than disorganized work. Problems for which just answers are shown, without the work being shown, will receive ZERO credit!

4. The setup (PHYSICS) of a problem counts more than the mathematics of working it out.

5. PLEASE write neatly. Before handing in the solutions, PLEASE: a) put problem solutions in numerical order, b) number the pages & put them in order, & c) clearly mark your answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves.

6. NOTE!! The words “DISCUSS” & “EXPLAIN” below mean to write English sentences in the answer. They DON’T mean to answer using only symbols. Answers to such questions containing only symbols without explanation of what they mean will get ZERO CREDIT!!! It would also be nice if graduate physics students would try to write complete, grammatically correct English sentences!

NOTE: I HAVE 10 EXAMS TO GRADE!!! PLEASE HELP ME GRADE THEM EFFICIENTLY BY FOLLOWING THESE

SIMPLE INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY RESULT IN A LOWER GRADE!! THANK YOU!!

NOTE!!!! YOU MUST ANSWER QUESTION 1! ANSWER ANY 3 OF THE OTHERS!

So, answer 4 questions total. Each is equally weighted & worth 25 points for 100 points total.

1. THIS QUESTION IS REQUIRED!!

a. Briefly DISCUSS, using WORDS, with as few mathematical symbols as possible, the PHYSICAL MEANINGS of the following terms:1) Microcanonical Distribution, 2) Canonical Distribution, 3) Grand Canonical Distribution, 4) Fermi Energy, 5) Pauli Exclusion Principle, 6) Classical Statistics, 7) Quantum Statistics, 8) Equipartition Theorem.

b. Briefly DISCUSS, using WORDS, NOT symbols, the fundamental differences between Fermions & Bosons and how these differences lead to the fundamentally very different Fermi-Dirac & Bose-Einstein Statistics. (That is, what are the basic, intrinsic properties that distinguish Fermions & Bosons?) In this discussion, be sure to mention the many particle wavefunctions for both kinds of systems & include the qualitative differences expected between the many particle ground states of Fermi-Dirac & Bose-Einstein systems.

c. In class, we discussed two different models to calculate the contribution of lattice vibrations to the heat capacity at constant volume, Cv, of a solid. In Ch. 7, we first discussed the Einstein Model. Then (for a couple of lectures) we wnet forward to Ch. 10 to discuss the Debye model. Briefly DISCUSS, using WORDS, NOT symbols, the major differences between the Einstein & Debye Models. Which model gives a theoretical temperature dependence for the low temperature Cv which is in agreement with experiment?

NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!

2. A solid contains N identical, non-interacting atoms. Each has a nucleus with spin S = 1. According to quantum mechanics, each nucleus can therefore be in any one of 3 quantum states, labeled by quantum number m which can have the values m = 1, 0, -1. This quantum number is a measure of the projection of the nuclear spin along a crystal axis of the solid. The electric charge distribution of a nucleus is not spherically symmetric, but ellipsoidal, so the energy of a nucleus depends on it’s spin orientation with respect to the internal electric field at it’s location. In the 3 spin states, the energies of a nucleus are εm = ε for m = 1 and m = -1 and is εm = 0 for m = 0.

a. Find an expression for the nuclear contribution to the partition function Z for this solid. Don’t forget the Gibbs correction!

b. Calculate the nuclear contribution to the mean energy for this solid.

c. Calculate the nuclear contribution to the heat capacity C for this solid

d. Calculate the nuclear contribution to the entropy S for this solid.

NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!

3. Consider N identical, non-interacting magnetic atoms in a solid at thermal equilibrium at temperature T. The solid is in a static external magnetic field H. Assume that H is in the z direction. Each atom has a magnetic moment μ. Use CLASSICAL statistical mechanics to find the CLASSICAL expressions for the thermodynamic properties asked for in the following. Hints: You need to use the classical energy of a magnetic moment μ in a magnetic field H: E = -μH cosθ. θ is the angle between μ & H. The results of parts a,b,c will be different than the quantum mechanical results from Reif’s Ch. 7 that we discussed in class.

a. Find an expression for the partition function Z for this system. Don’t forget the Gibbs

correction!

b. Calculate the mean z component of magnetic moment and the mean z

component of magnetization .

c. Calculate the mean energy and the heat capacity C for this system.

d. Calculate the entropy S for this system.

NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!

4. The natural logarithm of the partition function z for a single one-dimensional, quantum mechanical, simple harmonic oscillator of frequency ω in thermal equilibrium at temperature T was shown in Ch. 7 to be: ln(z) = - (½)βħω – ln(1 – e-βħω) where β = (kBT)-1.

Consider system of N such oscillators. They are identical and non-interacting.

a. Find an expression for the partition function Z for this system. Don’t forget the Gibbs correction!

b. Calculate the mean energy for this system. DISCUSS the physical meaning of the two terms in the mean energy. Calculate the heat capacity C of this system.

c. Calculate the entropy S of this system.

d. Evaluate the heat capacity of part b in the limit of high temperatures (ħω > kBT). DISCUSS the physical meaning of this limit.

NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!

5. Consider a classical monatomic, NON-IDEAL gas with N particles confined to volume V which is in thermal equilibrium at temperature T. When discussing Ch. 10 of Reif, we showed that the natural logarithm of the partition function Z for this system has the form:

ln(Z) = (3/2)N ln(A/β) + ln(ZU), where A = (2πm/h2), β = (kBT)-1,

and ln(ZU) has the approximate form: ln(ZU) = N ln(V) - (N2/V)B2(T)

Here B2(T) is called the “2nd Virial Coefficient”. It is a complicated integral which depends

on the form of the interaction potential between two particles. We also showed that this Z

gives an approximate equation of state for this gas of the form: P = kBT[n + B2(T)n2]. Here, n = (N/V) is the number density. Of course, there are other important properties of a

gas besides it’s equation of state. For example:

a. Derive expression for the mean energy of this gas.

b. Derive an expression for the heat capacity at constant volume, Cv of this gas.

c. Derive an expression for the entropy S of this gas.

d. Derive an expression for the chemical μ potential of this gas.

Note: For each part, assume that B2(T) is a known function & express your answers in general in terms of it and it’s temperature derivatives.

NOTE!!!! WORK ANY 3 OUT OF PROBLEMS 2 through 6!

6. A gas of N non-interacting hydrogen molecules (H2 molecules) is in thermal equilibrium at absolute temperature T.

a. Assume that, for calculating vibrational properties, each H2 molecule can be treated as a quantum mechanical simple harmonic oscillator with natural frequency ω. Find an expression for the vibrational partition function Zvib of this gas.

b. Assume that, for calculating rotational properties, an H2 molecule can be treated as a quantum mechanical rigid rotator. Thus, the quantized rotational energy states have energies of the form EJ = J(J+1)(ħ)2/I where J is the rotational quantum number and I is the moment of inertia, for which you may assume a classical “dumbbell” model. Recall from quantum mechanics that, in addition to the quantum number J, each rotational energy state is also characterized by a quantum number m, which can have any of the 2J +1 values m = -J, -(J - 1), -(J - 2),…,….(J - 2), (J - 1), J. So, each rotational energy EJ is (2J + 1)-fold degenerate. Of course, this degeneracy must be accounted for when the partition function is calculated. Write a formal expression (“formal expression” means leave it as a sum or an integral which can’t easily be evaluated in closed form) for the rotational partition function Zrot of this gas. Evaluate it in the high temperature limit. What does “high temperature limit” mean?

c. Still in the high temperature limit, calculate the total mean energy, including translational, vibrational, and rotational parts.

d. Calculate the specific heat at low temperatures, assuming that the temperature is still high enough that the N H2 molecules remain in a gaseous form.

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